Received January 2014.

The authors acknowledge the support of the Power Systems Engineering Research Center (PSerc) which is a Generation III National Science Foundation Industry University Cooperative Research Center, awards EEC-0001880, EEC-0968993. The authors also acknowledge the support

of the U.S. Department of Energy for its PSerc future grid initiative.

The authors are with the School of Electrical, Computer, and Energy Engineering at Arizona State University, Tempe, AZ 85287, and can be

reached at {brian.pierre, heydt}@asu.edu.

Brian J. Pierre

Gerald T. Heydt

Arizona State University

Tempe, Arizona USA

Abstract High phase order (HPO) systems have been proposed at the early inception of power

transmission engineering, but few direct applications have been made. High phase order trans-

mission should be considered as an alternative in the case of high power density applications. In

this paper an analysis of transposition of HPO overhead transmission lines is presented and volt-

age unbalance in HPO systems is considered. Definitions are presented for ‘fully transposed’ and

‘roll transposed’ along with advantages and disadvantages of each. A generalized voltage unbal-

ance factor (VUF) is introduced and utilized to determine the benefits of transposition. The gen-

eralized VUF is compared with three other possible unbalance factors to determine if the gener-

alized VUF is an appropriate indication of unbalance. Exemplary results are presented for six

phase and twelve phase designs. Conclusions show that the generalized VUF is a good indication

of transmission line voltage unbalance and certain configurations may not need full rotation

transposition to minimize the unbalance factor. The transposition analysis and voltage unbalance

are considerations in the assessment of HPO as a high power transmission alternative.

Keywords Power transmission, transmission engineering, high phase order, six phase, transposi-

tion, voltage unbalance factor.

Transposition and Voltage Unbalance in High

Phase Order Power Transmission Systems

1. Introduction

This paper relates to high phase order (nφ > 3) power transmission with a focus on transposition and

voltage unbalance. Although there are presently no bulk power applications of HPO systems, there have

been demonstration studies [1-5] and various other studies [6-11]. It is also possible that high phase order

loads (that is loads served directly at high phase order, e.g., induction motors [12-14]) could be served

directly from HPO distribution circuits. It is postulated that when considering all alternatives for bulk

power transmission, innovative concepts such as high phase order should be included. Limited right of

way, economic issues, and environmental issues also come into play in transmission engineering, and

some of these factors may favor high phase order. There are many advantages and disadvantages to HPO

designs; these are not discussed in this paper, but can be found in the literature, e.g. [15]. This paper will

focus on HPO transposition and voltage unbalance.

Transposition of overhead AC transmission circuits has been used for decades. Transposing three

phase lines has been studied extensively, e.g., [16, 17]. However, transposing HPO lines is infrequently

discussed. The reasons for transposing lines can be found in the literature, e.g. [18-20]. In brief, untrans-

posed transmission lines cause voltage and current unbalance that lead to problematic operating condi-

tions, counter rotating torque on machines, increased losses in some cases, and protection considerations.

For analysis purposes this paper focuses only on unbalance caused by the transmission line (i.e., self im-

pedances and mutual coupling among the phases).

The term ‘fully transposed’ may take on a range of meanings. For present purposes, assume that a ful-

ly transposed transmission line has a line impedance matrix in the form,

𝑍𝑝ℎ = [

𝑆 𝑀𝑀 𝑆

⋯ 𝑀⋮

⋮𝑀 ⋯

⋱ 𝑀𝑀 𝑆

], (1)

where, S is the self impedance, and M is j times the mutual reactance between phases. It may be possible

to include resistance in the mutual terms, but this complication is omitted. The shunt capacitance is readi-

ly modeled through the use of a capacitance matrix: the capacitance matrix is in the same form as (1)

with the capacitance to ground on the diagonals and the phase-phase capacitance on the off-diagonals. For

the fully transposed case, the off-diagonal entries of the capacitance matrix are all equal and the diagonal

entries are all equal. In (1), Zph is defined in terms of the line terminal voltages V1 and V2, and the line

current I. For an n phase circuit,

V1 – V2 = ZphI.

The V and I vectors are complex n-vectors, and the Zph matrix is a complex, symmetrical n by n matrix.

The term ‘roll transposed’ refers to a transmission line that is physically ‘rolled’ as pictorially illus-

trated for a six phase example in Fig. 1. Fig. 1 may be in any configuration and any number of phases, the

same roll transposition method applies to render the capacitance matrix and impedance matrix in the n by

n symmetrical circulant Toeplitz (SCT) form,

𝑍𝑝ℎ =

[

𝑆 𝑀1 𝑀2 … 𝑀𝑛/2 … 𝑀2 𝑀1𝑀1 𝑆 𝑀1 ⋱ ⋱ 𝑀2𝑀2 𝑀1 𝑆 ⋱ ⋮

⋮ ⋱ ⋱ ⋱ 𝑀𝑛/2𝑀𝑛/2 ⋮

⋮ ⋱ 𝑀2𝑀2 𝑀1𝑀1 𝑀2 … 𝑀𝑛/2 … 𝑀2 𝑀1 𝑆 ]

(2)

for even n. For odd n, Zph is in the n by n SCT form,

𝑍𝑝ℎ =

[

𝑆 𝑀1 𝑀2 … 𝑀(𝑛−1)/2 𝑀(𝑛−1)/2 … 𝑀2 𝑀1𝑀1 𝑆 𝑀1 ⋯ … 𝑀(𝑛−1)/2 ⋱ … 𝑀2𝑀2 𝑀1 𝑆 ⋱ ⋱ ⋮

⋮ ⋱ ⋱ ⋱ 𝑀(𝑛−1)/2𝑀(𝑛−1)/2 … 𝑀(𝑛−1)/2𝑀(𝑛−1)/2 ⋱ ⋮

⋮ ⋱ 𝑀2𝑀2 𝑀1𝑀1 𝑀2 … 𝑀(𝑛−1)/2 𝑀(𝑛−1)/2 … 𝑀2 𝑀1 𝑆 ]

, (3)

where Mi is j times the mutual reactance determined by the distance between the corresponding two con-

ductors, di, as shown in Fig. 2. In Fig. 2, only circularly symmetric configurations are shown, i.e., the

conductors are located on a circumscribed circle at points 360/n degrees apart. Circulant matrices have

special properties that are discussed in [21, 22]. For three phase systems, roll transposed conductors are

fully transposed; however, for HPO systems this observation is not true.

Fig. 1 An illustration of ‘rolling’ the phases of a six phase transmission line in a transposition process

Fig. 2 Circularly symmetric transmission line conductor configuration examples for n = 3, 6, 12

2. A comparison of fully transposed vs. roll transposed transmission lines

The fault analysis for a fully transposed line is straightforward because all the sequence impedances

are equal (other than the zero sequence impedance). This property results in relatively simple protection

schemes for fewer fault types as compared to the rolled transposition case. However, there are disad-

vantages to fully transposing an HPO transmission line.

HPO transmission lines have lower phase to phase voltage than comparable three phase counterparts.

This lower voltage permits closer phase conductor spacing. The closer phase spacing results in potential

benefits [15]. If HPO lines were fully transposed, the phase compaction benefits may be lost: this is due

to the fact that greater phase spacing is needed between phases A and C than A and B (for example) in

HPO lines. For this reason high phase order transmission lines should be roll transposed, not fully trans-

posed. In addition, roll transposed designs are relatively simple from a construction point of view, where-

as fully transposed designs require exchanging conductors and different tower configurations. Perhaps

the most salient reason that roll transposition is preferred over full transposition is that the number of

transposition sections needed is dramatically reduced in the case of roll transposition. Table I shows the

number of transposition sections required for differently configured transmission lines to be either roll

transposed or fully transposed.

In Table I the configuration of most interest is the circularly configured case because this design

makes the best use of higher power transfer with limited right of way (ROW). Double vertical or vertical-

ly configured HPO transmission lines are also of interest because they can transfer high power with short-

er tower heights as compared to comparable multicircuit three phase designs.

TABLE I CONDUCTOR CONFIGURATIONS FOR TRANSPOSITION

SECTIONS FOR FULL TRANSPOSITION (n = NUMBER OF PHASES)

HPO conductor configuration

(examples shown)

Transposition sec-

tions needed to be

‘roll transposed’

Transposition sec-

tions needed to be

‘full transposed’

Symmetrical circular

(equal spacing)

n

𝑛!

2

Vertical

(equal spacing)

𝑛!

Double vertical

(equal spacing)

𝑛!

2

Horizontal

(equal spacing) 𝑛!

2

Arbitrary

𝑛!

3. Voltage unbalance factor

A commonly accepted unbalance factor for three phase voltages is,

𝑈3𝑝ℎ =|𝑉−|

|𝑉+| (4)

with V– and V+

denoting the negative and positive sequence voltages. Equation (4) will be termed the

‘three phase voltage unbalance factor’ for purposes of this paper. Other common voltage unbalance fac-

tors are defined by NEMA and IEEE [23] and further unbalance factors are discussed in [18, 20]. For nφ

circuits there are nφ sequence voltages. Certain other sequence voltages that are similar to negative se-

quence voltage in the three phase case may result in undesired torque in rotating machines. For this rea-

son, it appears to be useful to define an nφ VUF that models the positive sequence voltage in the denomi-

nator and all other sequences in the numerator. Therefore, a ‘generalized voltage unbalance factor’ is in-

troduced,

𝑈𝑛𝑝ℎ =√∑ |𝑉𝑞|

2𝑛−1𝑞=2 + |𝑉0|

2

|𝑉1|, (5)

where n is the number of phases, zero sequence voltage is V0, positive sequence voltage is V1, negative

sequence voltage is Vn-1 and the other sequences 1< q < n-1 are sequences that do not exist in the three

phase case. To measure unbalance for an HPO transmission line the generalized VUF will be compared

with three other possible measures of VUF. In this comparison it will be shown that the generalized VUF

is a good indication of unbalance for an HPO transmission line.

The generalized VUF is used in this paper to illustrate the consequences of transposition and geome-

try of the conductors for HPO transmission lines. A possible use of HPO is to supply HPO motors direct-

ly. Therefore, an HPO complex voltage unbalance factor (CVUF) could be useful to determine the derat-

ing of HPO motors with voltage unbalance. CVUFs are discussed in the literature [24, 25]. However, it

should be noted that even a CVUF may not be a sufficient indication of voltage unbalance [26, 27].

4. Voltage unbalance factor as a measure of transposition effectiveness

Transposition is needed in long line designs. However, transposition is expensive and can cause stress

on conductors (e.g., mile for mile, faults are more likely in transposition sections than ordinary sections

[28], and mechanical stress occurs in transposition sections). For these reasons, it is beneficial to deter-

mine the number of transpositions needed to minimize the VUF to an acceptable level. If the transmission

line is completely roll transposed, the VUF due to the transmission line impedances is zero.

The four VUFs compared are calculated in a way described in the flow chart in Fig. 3. For phase or-

der n = 3k where k = 1, 2, …, a way to measure voltage unbalance for an HPO transmission line is to uti-

lize the voltage unbalance of each three phase circuit that may be derived from the k different three phase

subcircuits. For example, for a six-phase transmission line, measurements of unbalance would be taken

from two three phase circuits, i.e., unbalance on phases A, C, E, and unbalance on phases B, D, F. Both

the maximum unbalance and the average unbalance are analyzed and compared to the generalized VUF.

Fig. 3 Flow chart to calculate four unbalance factors for comparison purposes

A VUF comparison will be made for the following test bed example, however note that similar exam-

ples follow the same trends and results. A long line example is used to emphasize the impact of transposi-

tion on voltage unbalance. Consider a 200 mile long six phase 79.7 kV voltage line-line (Vll) transmission

line and a twelve phase 41.2 kV Vll transmission line; this will be equivalent to a three phase 138 kV Vll

transmission line. All example lines will carry 100 MVA of power at 0.8 power factor. Drake conductors

will be utilized with 5 ft. phase to phase spacing. The height of the lowest conductor is 35 feet. Examples

will be shown for six phase and twelve phase designs in configurations shown in Table II [15]. Dashed

lines are insulators. The VUFs are calculated for each number of transposition sections applied to the ex-

ample transmission lines, and the VUFs are compared in Figures 4-6.

Table II Different HPO configurations for the cited test bed example

(Drake, 200 mile, 100 MVA line)

Configuration A

six phase circular

Configuration B

six phase double

vertical

Configuration C

six phase verti-

cal

Configuration D

twelve phase

circular

Configuration E

twelve phase

double vertical

Configuration D: Twelve phase circular

Notice how low the VUF is to begin with. For

this reason it is possible that no transposition

is needed for circular designs.

Notice the generalized VUF is similar to the

avg. of the four individual three phase circuits

VUF and the maximum of the four individual

three phase circuits VUF.

Notice how the unbalance factor for 3 phase

applied to 12 phase is not a good indication of

unbalance (solid line).

Configuration A is very similar to this case,

and therefore not shown.

Fig. 4 VUF vs. number of transpositions sections for Configuration D (from Table II)

Configuration C: Six phase vertical

Notice the generalized unbalance factor is

similar to the average of the two individual

three phase circuits unbalance factor and the

maximum of the two individual three phase

circuits unbalance factor.

Fig. 5 VUF vs. number of transpositions sections for Configuration C (from Table II).

Configuration E: twelve phase double vertical

Notice that after n/2 transposition the unbal-

ance is the second lowest.

Notice how the generalized VUF is similar to

the other measurements of VUF. Again show-

ing that the generalized VUF is a good indica-

tor of voltage unbalance.

Configuration B is very similar to this case,

and therefore not shown.

Fig. 6 VUF vs. number of transpositions sections for Configuration E (from Table II).

Note that results were also checked with NEMA and IEEE voltage unbalance factors, and similar

results were verified. The generalized VUF followed the same trends and similar magnitudes as the NE-

MA and IEEE VUFs. A few key results can be drawn from Figs. 4-6:

A. From these results it can be shown that generally for the first n/2 transpositions, with increase in

transposition sections there is a decrease in VUF. Therefore one can determine if all transpositions

are needed, or if it is more economical to use just enough transposition sections to reach an ac-

ceptable unbalance.

B. Notice that if the transmission line is circularly configured the VUF is very small and may not need

any transposition.

C. Notice if the transmission line is double vertically configured, after n/2 transpositions, the VUF is

often the second lowest. For this reason, it may be beneficial to use n/2 roll transposition sections

instead of fully rotating the conductors and using n roll transpositions.

D. Notice that the defined generalized VUF is similar and corresponds to the average of the multicir-

cuit three phase VUF and the maximum of the multicircuit three phase VUF. The generalized VUF

is also shown to be a better indicator than using the three phase VUF applied to the HPO case.

Therefore, an appropriate indication of voltage unbalance can be obtained with (5) without addi-

tional steps of converting to multicircuit three phase.

E. If a 3k order circuit (k = 1, 2, …) energizes k three-phase rotating loads, the maximum of the mul-

ticircuit three phase VUF will give a measure of the greatest impact on the three phase rotating

loads [29, 30] and rectifier loads [31]. The generalized VUF also affords an appropriate impact

measure. Further studies may also research a CVUF to give a better indication of the effects of

voltage unbalance on rotating loads [24, 25].

5. Conclusions

Although there are no HPO transmission lines in existence, it is beneficial to study the possibility of

HPO in the design of high density power transmission. This paper discusses two aspects of HPO trans-

mission: transposition and voltage unbalance. Definitions of ‘roll transposed’ and ‘fully transposed’ have

been presented with advantages and disadvantages of each. The conclusion is that roll transposition

should be used with HPO transmission lines that have high unbalance. In addition, a generalized voltage

unbalance factor was introduced and verified through comparison to other unbalance measures. The gen-

eralized unbalance factor is an appropriate indication of voltage unbalance for an HPO transmission line

and a better indication than using the traditional three phase unbalance factor. Fully rotating the phases in

roll transposition may not be needed, that is, it may be sufficient to use just enough transposition sections

to minimize the unbalance factor to an acceptable level. This is especially true for double vertical con-

structions where n/2 transposition sections may sufficiently attain a low level of voltage unbalance.

References

[1] M. Brown, R. Rebbapragada, T. Dorazio, J. Stewart, “Utility system demonstration of six phase pow-

er transmission,” Proc. IEEE Transmission and Distribution Conference, Dallas, TX, 1991.

[2] New York State Electric & Gas Corporation, “High phase order transmission demonstration: Final

report,” Empire State Electric Energy Research Corporation, Albany NY, 1997.

[3] J. Stewart, L. Oppel, G. Thomann, T. Dorazio, M. Brown, “Insulation coordination, environmental,

and system analysis of existing double circuit line reconfigured to six-phase operation,” IEEE Trans-

actions on Power Delivery, Vol. 7, No. 3, pp. 1628-1633, 1992.

[4] A. Apostolov, R. Raffensperger, “Relay protection operation for faults on NYSEG’s six-phase trans-

mission line,” IEEE Transactions on Power Delivery, Vol. 11, No. 1, pp. 191-196, 1996.

[5] R. Rebbapragada, H. Panke, H. Pierce, J. Stewart, L Oppel, “Selection and application of relay protec-

tion for six phase demonstration project,” IEEE Transactions on Power Delivery, Vol. 7, No. 4, pp.

1900-1911, 1992.

[6] H. C. Barnes, L. O. Barthold, "High phase order power transmission," CIGRE SC 32, Electra, No. 24,

1973.

[7] T. Dorazio, “High phase order transmission,” Proc. Southern Tier Technical Conference, Bing-

hamton, NY, 1990.

[8] T. Landers, R. Richeda, E. Krizanskas, J. Stewart, R. Brown. “High phase order economics: construct-

ing a new transmission line,” IEEE Transactions on Power Delivery, Vol. 13, No. 4, pp. 1521-1526,

1998.

[9] N. Bhatt, S. Venkata, W. Guyker, W. Booth, “Six-phase (multi-phase) power transmission systems:

Fault analysis,” IEEE Transactions on Power Apparatus and Systems, Vol. 96, No. 3, pp. 758-767,

1977.

http://ieeexplore.ieee.org.ezproxy1.lib.asu.edu/xpl/articleDetails.jsp?tp=&arnumber=714849&contentType=Journals+%26+Magazines&searchField%3DSearch_All%26queryText%3Dsix+phase+transmission+powerhttp://ieeexplore.ieee.org.ezproxy1.lib.asu.edu/xpl/articleDetails.jsp?tp=&arnumber=714849&contentType=Journals+%26+Magazines&searchField%3DSearch_All%26queryText%3Dsix+phase+transmission+power

[10] W. Abu-Elhaija, F. Amoura, “Fault analysis of six phase power system using six phase symmetrical

components,” Electric Power Components and Systems, Vol. 33, No. 6, pp. 657-671, 2005.

[11] M. Farghaly, E. Ibrahim, “An important study of the transverse faults on twelve-phase power sys-

tems,” Electric Power Components and Systems, Vol. 34, No. 4, pp. 369-383, 2006.

[12] E. A. Klingshirn, “High phase order induction motors part I- description and theoretical considera-

tions,” IEEE Transactions on Power Apparatus and Systems, Vol. 102, No. 1, pp. 47 – 53, 1983.

[13] E. E. Ward, H. Härer, “Preliminary investigation of an inverter fed 5-phase induction motor,” Proc.

IEE, Vol. 116(B), No. 6, pp. 980 – 984, 1969.

[14] A. Iqbal, G. Singh, V. Pant, “Steady-state modeling and analysis of six-phase synchronous motor,”

Systems Science & Control Engineering, Vol. 2, No. 1, pp. 236-249, 2014.

[15] J. Bortnik, ‘Transmission line compaction using high phase order transmission,’ doctoral thesis,

University of the Witwatersrand, Johannesburg, South Africa, 1998.

[16] E. Gross, S. Nelson, “Electromagnetic unbalance of untransposed transmission lines,” IEEE Trans-

actions on Power Apparatus and Systems, Vol. 74, No. 3, pp. 887-893, 1955.

[17] J. Ma, S. Fortin, F. Dawalibi, “Analysis and mitigation of current unbalance due to induction in

heavily loaded multicircuit power lines,” IEEE Transactions on Power Delivery, Vol. 19, No. 3, pp.

1378-1383, 2004.

[18] J. Mooney, “Economic analysis and justification for transmission line transposition,” Proc. IEEE

Transmission and Distribution Conference and Exposition, New Orleans, 2010.

[19] M. Hesse, “Circulating currents in paralleled untransposed multicircuit lines: I- numerical evalua-

tions,” IEEE Transactions on Power Apparatus and Systems, Vol. 85, No. 7, pp. 802-811, 1966.

[20] T. Chen, “Evaluation of line loss under load unbalance using the complex unbalance factor,” IEEE

Proc. Generation, Transmission and Distribution, Vol. 142, No. 2, pp. 173-178, 1995.

[21] G. Heydt, B. Pierre, “Integrating transmission and distribution engineering eventualities,” Proc.

PSERC Future Grid Initiative, pp. 63-68, Madison, WI, 2013.

[22] R. Gray, Toeplitz and circulant matrices: A review, Foundations and Trends in Communications and

Information Theory, Now Publishers, Hanover MA, 2006.

[23] J. Kim, E. Lee, D. Lee, J. Lee, “Comparison of voltage unbalance factor by line and phase voltage,”

International Conference Proc. on Electrical Machines and Systems, Vol. 3, pp. 1998-2001, 2005.

[24] Y. Wang, “Analysis of effects of three-phase voltage unbalance on induction motors with emphasis

on the angle of the complex voltage unbalance factor,” IEEE Transactions on Energy Conversion,

Vol. 16, No. 3, pp. 270-275, 2001.

[25] C. Reineri, J. Gomez, N. Campetelli, “Adjustable speed drives supplied with unbalanced voltages:

Study under complex voltage unbalance factor parameter,” IEEE Proc. on Transmission and Distri-

bution Conference and Exposition, Bergamo, Italy, 2008.

[26] A. Filho, et al. “Analysis of the complex voltage unbalance factor behavior resulting from the varia-

tion of voltage magnitudes and angles,” IEEE International Conference Proc. on Harmonics and

Quality of Power, Bergamo, Italy, 2010.

[27] A. Filho, D. Garcia, F. Nascimento, J. Angarita, “Study of voltage unbalance conditions based on the

behavior of the complex voltage unbalance factor (CVUF),” IEEE Proc. Transmission and Distribu-

tion Conference and Exposition, Sao Paulo, Brazil, pp. 184 – 189, 2010.

[28] V. Subba Rao, “Balancing bundle conductor transmission line constants without transpositions,”

Proc. of the Institution of Electrical Engineers, Vol. 112, No. 5, pp. 931 – 940, 1965.

[29] C-Y. Lee, “Effects of unbalanced voltage on the operation of a three-phase induction motor,” IEEE

Trans. on Energy Conversion, Vol. 14, No. 2, pp. 202-208, 1999.

[30] R. F. Woll, “Effect of unbalanced voltage on the operation of polyphase induction motors,” IEEE

Trans. on Industry Applications, Vol. 11, No. 1, pp. 38 – 42, 1975.

[31] K. Olejniczak, G. Heydt, “Basic mechanisms of generation and flow of harmonic signals in balanced

and unbalanced three-phase systems,” IEEE Trans. on Power Delivery, Vol. 4, No. 4, pp. 2162 –

2170, 1989.